The fact that the massless version of lower fermions have a vanishing Hamiltonian and the upper massless fermions have a non-vanishing one and they behave in very different ways can also be traced back to the representation theory of the Carroll algebra, where the Hamiltonian is a central element and the two cases \(H=0\)  and \(H\ne 0\)  leads to different classes of representations. See [1]} for more details on these classes of representations.
